Stochastic Differential Equations

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.

Everywhere in this chapter we use the notions and notation introduced in Chapter 5. We describe a stochastic differential equation of a special sort on those groups of diffeomorphisms that arise in the applications to viscous hydrodynamics described in Section 16.4 below (see, e.g., Gliklikh (1989, 1994, 2003)). This class of equations is characterized by the fact that they involve finite-dimensional Wiener processes. It should be pointed out that a theory of stochastic differential equations on infinite-dimensional manifolds involving infinite-dimensional Wiener processes exists (see, e.g., Belopolskaya and Dalecky (1989), Brzezniak and Elworthy (2000), Elworthy (1982)) and is used in viscous hydrodynamics (see, e.g., Cipriano and Cruzeiro (2007)). However, the description of this theory requires a complicated functional-analytic machinery that is not included in our exposition. For simplicity of presentation, we restrict ourselves to the finite-dimensional version of the theory since, in applications, the theories yield very similar results.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

This paper uses stochastic differential equation and risk theory to study the probability model of insurance company's bankruptcy function and the proportion model of financial insurance investment. It also studies the application of stochastic probability to the investment decision-making of institutional investors. Taking the four major insurance industries in China as an example, it gives the strategy of venture capital.

The theory of stochastic differential equation on the field of $p$-adics is initiated by Kochubei. In this article, we will focus on a class of random walks with a certain integrability to develop the theory of stochastic analysis in a way similar to the existing theory of stochastic analysis based on the Brownian motion. In fact, for any random walk in the class, we can introduce the notion of stochastic integral with respect to the random walk and justify the existence of the solution of the stochastic differential equations based on the random walk, where the stochastic differential equations admit not only Lipschitz continuous path dependent coefficients but also continuous coefficients growing at most linearly with respect to $p$-adic norm. Finally, we will see an example of stochastic process which can be covered by Dirichlet space theory and obtained also by solving stochastic differential equation.

Adaptive stepsize based on control theory for stochastic differential equations

Stochastic partial differential equations constitute a relatively new subject. There have been a number of interesting papers in the last few years, but the field is still in its infancy if compared with the highly developed theory of stochastic ordinary differential equations. In this paper we report on a study of a non linear heat equation in a finite interval of space subject to a white noise forcing term. Due to the non linearity the equation, without the forcing term, exhibits several equilibrium configurations two of which are stable, actually asymptotically stable. The solution of the complete forced equation is a stochastic process in space and time with continuous sample paths whose behaviour we study in the limit of small noise. We obtain lower and upper bounds for the probability of large fluctuations and then apply our estimates to the calculation of the transition probability between the stable configurations (tunneling).

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

This paper is focused on the development of a method to determine variances of the stochastic responses in transmission line models with random excitations. It is based on the theory of stochastic differential equations (SDE), namely a system of vector linear SDEs with additive noise is formulated to be prepared to simulate respective stochastic trajectories. One possibility to get the moments of the stochastic process is to determine sample statistics from a set of individual trajectories. This is, however, rather time consuming, and it's benefit when compared to common methods is not so obvious. In this paper differential equations for getting directly the first two stochastic response moments of transmission line models under randomly varying excitations are formulated and solved. A verification is performed via the above stated approach, while the Student's t-distribution can directly be used to determine respective confidence intervals in case of an additive noise. All simulations have been performed in a Matlab language environment.

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

In this article, we comment on the well known connection between noise and Q-Wiener processes. In physical applications, noise is usually given as a generalized Gaussian process and all assumptions are formulated in terms of the correlation functional. In contrast, the mathematical theory of stochastic partial differential equations on bounded domains is in many cases formulated with Q-Wiener processes. Our aim is to relate frequently made assumptions on the covariance operator Q to assumptions on the correlation function of the noise process, which is the kernel of Q. One of our main results gives necessary and sufficient conditions when the Wiener process is given as a series expansion in terms of eigenfunctions of the Laplacian.

The paper deals with a technique for the analysis of multiconductor transmission lines (MTL) with randomly varied parameters, that is based on the theory of stochastic differential equations (SDE). Sets of stochastic trajectories are computed as voltage or current responses, accompanied by relevant sample means and confidence intervals. The MTL model is based on a cascade connection of generalized RLGC networks, terminating circuits are replaced by their generalized Thévenin equivalents. To develop model equations a state-variable method is applied, and then a corresponding vector SDE is formulated. Finally, a stochastic implicit Euler numerical technique is used for the numerical solution being consistent with Itô stochastic calculus. All the computation were done in the MATLAB <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">®</sup> language and deterministic responses are also stated via a numerical inverse Laplace transforms (NILT) procedure to verify the results.

Nonlinear mechanisms of long-term variations in the Caspian Sea level are described. It is shown that with account taken of the dependence of the evaporation depth from the Volga basin surface on soil moisture content and the dependence of the evaporation depth from the sea surface on its level, we obtain a fundamentally new (chaotic) oscillation mechanism with several attraction levels. The stochastic differential equations describing the water budget of the sea basin and the sea proper and the respective solutions of the Fokker–Planck–Kolmogorov equation are shown to have stationary bimodal density of the level probability. The random process, characterizing the sea level variations at a nonlinear dependence between the evaporation rate and the level is found to be non-Gaussian. Noise-induced transitions, caused by nonlinear evaporation processes are described. A new nonlinear stochastic theory describing the Caspian Sea level variations and based on predicted physical effects is suggested.

The tenth special issue of the Japanese Journal of Statistics and Data Science (JJSD) focuses on statistics for stochastic processes. With the advancement of information and communication technology and measurement technology, a vast amount of diverse time series data have been readily obtained. Consequently, the significance of stochastic processes as mathematical models for random phenomena evolving in time or space has increased dramatically, and statistics for various types of stochastic processes have been rapidly developing. In fact, a multitude of studies on statistics for stochastic processes such as ARMA, CARMA, COGARCH, diffusion processes, fractional Brownian motions, GARCH, jump-diffusion processes, Lèvy processes, Markov processes, point processes, random fields, stochastic differential equations and stochastic partial differential equations have been conducted. Nonetheless, because of the mathematical complexity of statistics for stochastic processes, numerous theoretical and computational challenges remain to be addressed. This special feature contains ten original papers contributed by leading mathematical statisticians and data scientists on current topics in statistics for stochastic processes. The first paper examines the efficient particle Markov chain Monte Carlo method. The second and third papers respectively address the estimation problems of Hawkes processes and Cox proportional excess hazard models. The fourth and fifth papers investigate parametric estimation for probability partial differential equation models. The latter five papers focus on statistical inference for stochastic differential equation models driven by various stochastic processes, including the Wiener process, fractional Brownian motion, β-stable process and Lèvy process. The ten papers in the special feature are summarized as follows: Awaya and Omori propose an efficient particle Markov chain Monte Carlo methodology for the rolling-window estimation of state space models. They provide theoretical justifications for their proposed method and demonstrate the efficiency of B Masayuki Uchida